Finite Group Databasebeta

← C31×#13426 #21470185

Subgroups — C31×#13426 #21470185

Order 7936.

512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2733C211yesmaximal, characteristic
2C2×C31×#355C212yesmaximal, complemented
3C31×#1888C212yesmaximal, complemented
4C31×#1857C211yesmaximal, characteristic, complemented
5C31×#1394C211yesmaximal, characteristic, complemented
6C2×C31×#124C2212yes—
7C31×#519C2211yescharacteristic
8C31×#428—21no—
9C31×#418—22no—
10C2×C31×#101C2212yes—
11C2×C31×#101C2211yescharacteristic
12C31×#394—21no—
13C31×#392—22no—
14C31×#391C2211yescharacteristic
15C31×#390—21no—
16C31×#388—22no—
17C31×#355—24no—
18C31×#339—21no—
19C31×#339—22no—
20C22×Q8×C31C2311yescharacteristic
21C22×D4×C31—21no—
22C22×D4×C31—21no—
23C22×D4×C31D412yes—
24C2×M16×C31—21no—
25C2×M16×C31—22no—
26C31×#124—22no—
27C31×#124—22no—
28C31×#124—22no—
29C31×#124—22no—
30C4×Q8×C31—41no—
31C4×Q8×C31—41no—
32C31×#117—22no—
33C31×#117—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C42×C31D412yes—
41C2×C42×C31D411yescharacteristic
42C31×#107—41no—
43C31×#106—41no—
44C31×#102—42no—
45C31×#102—42no—
46C31×#101—21no—
47C31×#101—22no—
48C31×#101—22no—
49C31×#101D412yescomplemented
50C31×#95—21no—
51C31×#95—22no—
52C24×C31—22no—
53C2×Q8×C31—41no—
54C2×Q8×C31—41no—
55C2×Q8×C31C2×D411yescharacteristic
56C2×Q8×C31#4111yescharacteristic
57C2×Q8×C31#4112yes—
58C2×D4×C31—22no—
59C2×D4×C31—22no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C2×D4×C31—81no—
64C2×D4×C31—81no—
65C2×D4×C31—82no—
66C2×D4×C31—81no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—22no—
70C22×C4×C31C2×D412yes—
71C22×C4×C31C2×D411yescharacteristic
72M16×C31—22no—
73M16×C31—41no—
74M16×C31—42no—
75M16×C31—42no—
76C2×C8×C31—41no—
77C2×C8×C31—42no—
78C4⋊C4×C31—22no—
79C4⋊C4×C31—22no—
80C4⋊C4×C31—21no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—81no—
84C22⋊C4×C31—44no—
85C22⋊C4×C31—41no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—82no—
91C42×C31—22no—
92C42×C31—22no—
93C42×C31—41no—
94C42×C31—42no—
95C42×C31—41no—
96#13426C3111yesmaximal, characteristic, direct product factor, Sylow
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31#12011yescharacteristic
105Q8×C31—41no—
106Q8×C31—41no—
107Q8×C31—42no—
108D4×C31—44no—
109D4×C31—82no—
110D4×C31—84no—
111D4×C31—84no—
112C2×C4×C31—21no—
113C2×C4×C31—22no—
114C2×C4×C31—21no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31#12112yes—
127C2×C4×C31#12111yescharacteristic
128C2×C4×C31#12312yes—
129C2×C4×C31#12411yescharacteristic
130C8×C31—42no—
131C8×C31—82no—
132#2733C2×C3111yescharacteristic
133C2×#355C2×C3112yescomplemented
134#1888C2×C3112yescomplemented
135#1857C2×C3111yescharacteristic, complemented
136#1394C2×C3111yescharacteristic, complemented
137C22×C31—21no—
138C22×C31—21no—
139C22×C31—82no—
140C22×C31—82no—
141C22×C31—82no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31#39411yescharacteristic
147C22×C31B(4,𝔽2)11yescharacteristic
148C22×C31#45811yescharacteristic
149C4×C31—21no—
150C4×C31—22no—
151C4×C31—22no—
152C4×C31—21no—
153C4×C31—42no—
154C4×C31—82no—
155C4×C31—81no—
156C4×C31—81no—
157C4×C31—81no—
158C2×#124C22×C3112yes—
159#519C22×C3111yescharacteristic
160#428—21no—
161#418—22no—
162C2×#101C22×C3112yes—
163C2×#101C22×C3111yescharacteristic
164#394—21no—
165#392—22no—
166#391C22×C3111yescharacteristic
167#390—21no—
168#388—22no—
169#355—24no—
170#339—21no—
171#339—22no—
172C2×C31—21no—
173C2×C31—21no—
174C2×C31—162no—
175C2×C31#204411yescharacteristic
176C2×C31#220211yescharacteristic
177C2×C31#220411yescharacteristic
178C22×Q8C23×C3111yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4D4×C3112yes—
182C2×M16—21no—
183C2×M16—22no—
184#124—22no—
185#124—22no—
186#124—22no—
187#124—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—22no—
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4D4×C3111yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42D4×C3112yes—
199C2×C42D4×C3111yescharacteristic
200#107—41no—
201#106—41no—
202#102—42no—
203#102—42no—
204#101—21no—
205#101—22no—
206#101—22no—
207#101D4×C3112yescomplemented
208#95—21no—
209#95—22no—
210C31#1342611yescharacteristic, direct product factor, Sylow
211C24—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8C2×D4×C3111yescharacteristic
215C2×Q8#41×C3111yescharacteristic
216C2×Q8#41×C3112yes—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—82no—
225C2×D4—81no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4C2×D4×C3112yes—
230C22×C4C2×D4×C3111yescharacteristic
231M16—22no—
232M16—41no—
233M16—42no—
234M16—42no—
235C2×C8—41no—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C22⋊C4—44no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—82no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23C31×#12011yescharacteristic
263Q8—41no—
264Q8—41no—
265Q8—42no—
266D4—44no—
267D4—82no—
268D4—84no—
269D4—84no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4C31×#12112yes—
285C2×C4C31×#12111yescharacteristic
286C2×C4C31×#12312yes—
287C2×C4C31×#12411yescharacteristic
288C8—42no—
289C8—82no—
290C22—21no—
291C22—21no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22C31×#39411yescharacteristic
300C22C31×B(4,𝔽2)11yescharacteristic
301C22C31×#45811yescharacteristic
302C4—21no—
303C4—22no—
304C4—22no—
305C4—21no—
306C4—42no—
307C4—82no—
308C4—81no—
309C4—81no—
310C4—81no—
311C2—21no—
312C2—21no—
313C2—162no—
314C2C31×#204411yescharacteristic
315C2C31×#220211yescharacteristic
316C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C31
prime radicalC31×#2733C2
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C31×#355C2semidirect
3. C31×#1888C2semidirect
4. C31×#1857C2semidirect
5. C31×#1394C2semidirect
49. C31×#101D4semidirect
96. #13426C31direct (N × H)
133. C2×#355C2×C31semidirect
134. #1888C2×C31semidirect
135. #1857C2×C31semidirect
136. #1394C2×C31semidirect
207. #101D4×C31semidirect
210. C31#13426direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#27332
C2×C31×#3552
C31×#18882
C31×#18572
C31×#13942
#1342631